1

AT 2402 VEHICLE DYNAMICS _ UNIT III NOTES

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

MULTI DEGREE FREEDOM SYSTEMS

Orthogonality Principle

It states that the principal nodes are orthogonal to each other. In other words, expansion

theorem applied to vibration problems indicates that any general motion (x) for n masses may be

broke into components each of which corresponds to a principal nod

obtaining the response of vibration systems and is called modal analysis.

Modal analysis and mention its applications

Modal analysis can be applied to discrete as well as continuous systems.

Mode shapes & orthogonal decompos

Mode shapes of a structure can be extracted without measuring a series of frequency response

functions by implementing proper orthogonal decomposition on the measured response data. If

the proper orthogonal decomposition is app

proper orthogonal mode converges to the true normal modes of the structure. The degree of

deviation of the other extracted proper orthogonal modes from the true normal modes of the

structure depends on the spatial resolution, which is determined by the number of response

measurement positions. The suggested procedure of applying the proper orthogonal

decomposition to the cross-spectral density functions can extract all of the normal modes

contained in the structural responses without suffering from the limitation of the number of

response measurement positions. Experimental data of a homogeneous free

non-homogeneous free-free beam were used to compare the proper orthogonal modes and the

structural normal modes, and the results strongly support the applicability of the proper orthogonal

decomposition to the structural vibration analysis.

Closed coupled system.

In fig, an n degree translational and rotational close coupled system is given

the following equations can be written:x=e

m1ë1+K1e1+ K2(e1- e2)=0

m1ë2+K2(e2 - e1)+ K3(e2 – e3)=0

miëi+Ki(ei – ei -1)+ Ki+1(ei – ei+1)=0

We can ignore damping for free vibration, as the studies, as the natural frequencies are not

significantly effected by presence of damping. The above equations can also be used directly for

torsion with appropriate alternation.

[M] { ë}+[K] {e}=0

[M] is a square mass matrix

[K] is a stiffness matrix

In above equation, [K] {e} defines local force u

equation in this form called stiffness method using the displacements also called displacement

26

It states that the principal nodes are orthogonal to each other. In other words, expansion

theorem applied to vibration problems indicates that any general motion (x) for n masses may be

broke into components each of which corresponds to a principal node. This forms the basis of

obtaining the response of vibration systems and is called modal analysis.

Modal analysis and mention its applications

Modal analysis can be applied to discrete as well as continuous systems.

Mode shapes & orthogonal decomposition in vibration analysis.

Mode shapes of a structure can be extracted without measuring a series of frequency response

functions by implementing proper orthogonal decomposition on the measured response data. If

the proper orthogonal decomposition is applied on the time responses of the structure, only one

proper orthogonal mode converges to the true normal modes of the structure. The degree of

deviation of the other extracted proper orthogonal modes from the true normal modes of the

the spatial resolution, which is determined by the number of response

measurement positions. The suggested procedure of applying the proper orthogonal

spectral density functions can extract all of the normal modes

e structural responses without suffering from the limitation of the number of

response measurement positions. Experimental data of a homogeneous free

free beam were used to compare the proper orthogonal modes and the

ructural normal modes, and the results strongly support the applicability of the proper orthogonal

decomposition to the structural vibration analysis.

In fig, an n degree translational and rotational close coupled system is given. For free vibration,

the following equations can be written:x=e

e3)=0

ei+1)=0

We can ignore damping for free vibration, as the studies, as the natural frequencies are not

significantly effected by presence of damping. The above equations can also be used directly for

torsion with appropriate alternation.

In above equation, [K] {e} defines local force under static condition .The method of writing

equation in this form called stiffness method using the displacements also called displacement

It states that the principal nodes are orthogonal to each other. In other words, expansion

theorem applied to vibration problems indicates that any general motion (x) for n masses may be

e. This forms the basis of

Mode shapes of a structure can be extracted without measuring a series of frequency response

functions by implementing proper orthogonal decomposition on the measured response data. If

lied on the time responses of the structure, only one

proper orthogonal mode converges to the true normal modes of the structure. The degree of

deviation of the other extracted proper orthogonal modes from the true normal modes of the

the spatial resolution, which is determined by the number of response

measurement positions. The suggested procedure of applying the proper orthogonal

spectral density functions can extract all of the normal modes

e structural responses without suffering from the limitation of the number of

response measurement positions. Experimental data of a homogeneous free-free beam and a

free beam were used to compare the proper orthogonal modes and the

ructural normal modes, and the results strongly support the applicability of the proper orthogonal

. For free vibration,

We can ignore damping for free vibration, as the studies, as the natural frequencies are not

significantly effected by presence of damping. The above equations can also be used directly for

nder static condition .The method of writing

equation in this form called stiffness method using the displacements also called displacement

method. In fact, for close coupled systems, the finite element method yields identical equations as

above.

For free vibrations, the above solutions of can written as

{x}={X} cos pt

Where {X} represents the amplitudes of all the masses and p is the natural frequency. These

above equations reduce to

{[K]-p2[M]] {X} = 0

The above equations known as eigen value problem in

the aid of computer for a fairly large no of masses.p2 is called as the characteristic value of

equation.

There will be n such values of an n degree system. For each of the eigen value, there exists a

corresponding eigen vector{x}, also called as a characteristic vector. This eigen vector will

represent the mode shape for a given frequency of the system.

Far coupled system:

In fig an n degree far coupled system is shown with masses m1,m2,….,mi, at station 1,2,…..,

respectively. For a freely vibrating beam, the only external load is the inertia load due to masses

m1,m2….. Following the influence coefficient procedure discussed two degree of freedom, we can

write x=e

α11m1ë1+ α12m2ë2+………..+

α21m1ë1+ α22m2ë2+………..+

-----------------------------------------------------

αi1m1ë1+ αi2m2ë2+………..+ αiimi

In arriving at the above equations, the forces are treated as unknowns and expressed in terms of

flexibility factor or influences coefficie

contrary to the method adopted for close coupled systems where displacement are considered

unknowns and the stiffness matrix setup.

Denoting [α] as the influences coefficient matrix, the above

[D] { ë} + [I] {e} = 0

{ ë}+ [α]-1{e} = 0

[D] is the dynamic matrix

27

method. In fact, for close coupled systems, the finite element method yields identical equations as

vibrations, the above solutions of can written as

Where {X} represents the amplitudes of all the masses and p is the natural frequency. These

The above equations known as eigen value problem in matrix algebra and it can be solved with

the aid of computer for a fairly large no of masses.p2 is called as the characteristic value of

There will be n such values of an n degree system. For each of the eigen value, there exists a

eigen vector{x}, also called as a characteristic vector. This eigen vector will

represent the mode shape for a given frequency of the system.

In fig an n degree far coupled system is shown with masses m1,m2,….,mi, at station 1,2,…..,

respectively. For a freely vibrating beam, the only external load is the inertia load due to masses

m1,m2….. Following the influence coefficient procedure discussed two degree of freedom, we can

ë2+………..+ α1imiëi+e1=0

m2ë2+………..+ α2imiëi+e2=0

-----------------------------------------------------

αiimiëi+ei=0

In arriving at the above equations, the forces are treated as unknowns and expressed in terms of

flexibility factor or influences coefficients, hence this approach is called force or flexibility method,

contrary to the method adopted for close coupled systems where displacement are considered

unknowns and the stiffness matrix setup.

] as the influences coefficient matrix, the above equation can be written as

Or

method. In fact, for close coupled systems, the finite element method yields identical equations as

Where {X} represents the amplitudes of all the masses and p is the natural frequency. These

matrix algebra and it can be solved with

the aid of computer for a fairly large no of masses.p2 is called as the characteristic value of

There will be n such values of an n degree system. For each of the eigen value, there exists a

eigen vector{x}, also called as a characteristic vector. This eigen vector will

In fig an n degree far coupled system is shown with masses m1,m2,….,mi, at station 1,2,…..,i

respectively. For a freely vibrating beam, the only external load is the inertia load due to masses

m1,m2….. Following the influence coefficient procedure discussed two degree of freedom, we can

In arriving at the above equations, the forces are treated as unknowns and expressed in terms of

nts, hence this approach is called force or flexibility method,

contrary to the method adopted for close coupled systems where displacement are considered

equation can be written as

28

For a four degree of freedom , the natural frequencies are obtained by setting the determinant

equation to zero.

| [D] – [1/p2][I] | = 0

Orthogonality of mode shapes

The mode shapes of a dynamic system exhibit orthogonality property, which is very useful in

simplifying the analysis for forced and transient vibrations. Let [M] and [K] represent the mass and

stiffness properties of a system, expressed in the form

[K] {u} = λ2 [M] {u}

{u} is the state vector and λ2 is the eigen value. For each eigen value λi, there is a corresponding

mode shape. Consider any two modes say r and s. Then

[K] {ur} = λr2 [M] { ur }

[K] {us} = λs2 [M] { us}

Premultiply the first one by { us}T and the second one by { ur}T in the above equations. Then

{ us}T [K] {ur} = λr2 { us}T [M] { ur }

{ ur}T [K] {us} = λs2 { ur}T [M] { us}

{ us}T [M] {ur} = 0 r ≠ s

ģ ={ us}T [M] {ur}≠ 0 r=s

ģ = generalized mass of the system

{ us}T [K] {ur} = 0 r ≠ s

ķ={ us}T [M] {ur}≠ 0 r=s

ķ = generalized stiffness of the system

Modal analysis:

When the degree of freedom of the system is large and / or when the forcing functions are

nonperiodic, in such cases a more convenient method known as modal analysis can be used to

solve the problem. In this method, the expansion theorem is used, and the displacements of the

masses are expressed as a linear combination of the normal modes of the system.

29